Question

Let \(X\) and \(W\) be the working and subsequent repair times of a certain machine. Let \(Y=X+W\) and suppose that the joint probability density of \(X\) and \(Y\) is $$ f_{X, Y}(x, y)=\lambda^{2} e^{-\lambda y}, \quad 0

Short answer

(a) The pdf of X is: $$ f_X(x) = \lambda e^{-\lambda x}, \quad 0 < x < \infty $$ (b) The pdf of Y is: $$ f_Y(y) = \lambda^2 y e^{-\lambda y}, \quad 0 < y < \infty $$ (c) The joint pdf of X and W is: $$ f_{X, W}(x, w) = \lambda^2 e^{-\lambda (x+w)}, \quad 0 < x, w < \infty $$ (d) The pdf of W is: $$ f_W(w) = \lambda e^{-\lambda w}, \quad 0 < w < \infty $$

Step-by-step solution

(a) Finding the pdf of X

To find the pdf of X, we have to take the marginal of joint pdf f(x, y) over Y. We integrate f(x, y) over y, from x to infinity since the given range is 0 < x < y < ∞. $$ f_X(x) = \int_{x}^{\infty} \lambda^2 e^{-\lambda y} dy $$ Now, we find the integral: $$ f_X(x) = \left[ -\lambda e^{-\lambda y} \right]_{x}^{\infty} = \lambda e^{-\lambda x} $$ So, the pdf of X is: $$ f_X(x) = \lambda e^{-\lambda x}, \quad 0 < x < \infty $$

(b) Finding the pdf of Y

To find the pdf of Y, we have to take the marginal of joint pdf f(x, y) over X. We integrate f(x, y) over x, from 0 to y since the given range is 0 < x < y < ∞. $$ f_Y(y) = \int_{0}^{y} \lambda^2 e^{-\lambda y} dx $$ Now, we find the integral: $$ f_Y(y) = \left[ -\lambda^2 e^{-\lambda y}x \right]_{0}^{y} = \lambda^2 y e^{-\lambda y} $$ So, the pdf of Y is: $$ f_Y(y) = \lambda^2 y e^{-\lambda y}, \quad 0 < y < \infty $$

(c) Finding the joint pdf of X and W

We are given the joint pdf of X and Y but we need to find the joint pdf of X and W, where W = Y - X. We'll use the change of variables technique, so let's first find the Jacobian of the transformation. For the transformation: $$ Y = X + W \\ X = X $$ The Jacobian is: $$ J = \frac{\partial (x, y)}{\partial (x, w)} = \begin{vmatrix} \frac{\partial x}{\partial x} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial x} & \frac{\partial y}{\partial w} \end{vmatrix} = \begin{vmatrix} 1 & 0 \\ 1 & 1 \end{vmatrix} = 1 $$ Now we find the joint pdf of X and W: $$ f_{X, W}(x, w) = f_{X, Y}(x, x+w) |J| = \lambda^2 e^{-\lambda (x+w)} $$ So, the joint pdf of X and W is: $$ f_{X, W}(x, w) = \lambda^2 e^{-\lambda (x+w)}, \quad 0 < x, w < \infty $$

(d) Finding the pdf of W

To find the pdf of W, we can use the joint pdf of X and W that we just found and integrate out X. We integrate f(x, w) over x, from 0 to infinity. $$ f_W(w) = \int_{0}^{\infty} \lambda^2 e^{-\lambda (x+w)} dx $$ Now, we find the integral: $$ f_W(w) = \left[ -\lambda e^{-\lambda (x+w)} \right]_{0}^{\infty} = \lambda e^{-\lambda w} $$ So, the pdf of W is: $$ f_W(w) = \lambda e^{-\lambda w}, \quad 0 < w < \infty $$

Key concepts

Understanding Joint Probability Density

When studying the relationship between two continuous random variables, the joint probability density function (pdf) plays a crucial role. It provides information on the likelihood of both random variables simultaneously falling within a particular range. In technical terms, for two random variables X and Y, the joint pdf, denoted as f_{X, Y}(x, y), quantifies the probability that X is exactly x and Y is exactly y.

The joint pdf is a key instrument in the field of probability and statistics, as it serves as the foundation for determining the probability of events involving several random variables. The intuition behind it is that, while individual random variables give their own unique insights, it's their collective behavior, expressed by the joint pdf, that often interests statisticians and researchers.

Marginal Density Functions

To understand a single random variable within the context of a joint distribution, we utilize what is known as the marginal density function. Marginal density functions are derived from the joint probability density function but pertain to only one of the variables involved. They provide insight into the probability distribution of a random variable in isolation from others.

In the given exercise, the marginal density for each variable, X and Y, is obtained by integrating the joint pdf over the range of the other variable. For example, the marginal density function of X is found by integrating over all possible values of Y. This process essentially 'sums up' the joint probability over the dimension of Y, leaving us with a function that describes just the variable X—hence, it's 'marginally' depicting X independent of Y.

Integration in Probability

Integration is a fundamental operation in calculating probabilities for continuous random variables. In the context of probability density functions, integration allows us to find the probability that a random variable falls within a certain range. Essentially, it's the continuous analog of summing probabilities in a discrete setting.

When we integrate a probability density function over a range of values, we obtain the probability that the random variable lies within that interval. In our exercise, integration is used to transition from the joint pdf to marginal densities and to derive the probability densities of new random variables obtained through transformation. For students aiming to grasp these concepts, it's paramount to become comfortable with integration as it unlocks the ability to manipulate and understand various probability distributions.

Random Variables Transformation

The process of transforming random variables is a powerful technique used for simplifying complex stochastic processes or for deriving the distributions of new random variables. Transformation can involve re-expression or combination of existing random variables. When transforming variables, it's crucial to understand how these transformations affect their distributions.

In our exercise, the variable W is a transformation resulting from the subtraction of X from Y. To find the new joint pdf after this transformation, we compute the Jacobian, which provides the factor by which our function scales during the transformation. When it's equal to 1, like in our case, the form of the joint pdf remains unaffected by the scale of transformation and only requires substitution based on the new variables' relationships. Mastering variable transformation and the use of the Jacobian matrix is key for students venturing into higher-level statistics or any field that involves complex random behavior.